Car like robot how to park a car nonholonomic planning
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Car-Like Robot: How to Park a Car? (Nonholonomic Planning). Types of Path Constraints. Local constraints: e.g., lie in free space Differential constraints: e.g., have bounded curvature Global constraints: e.g., have minimal length. Types of Path Constraints.

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Car-Like Robot: How to Park a Car? (Nonholonomic Planning)

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Car like robot how to park a car nonholonomic planning

Car-Like Robot:How to Park a Car?(Nonholonomic Planning)


Types of path constraints

Types of Path Constraints

  • Local constraints: e.g., lie in free space

  • Differential constraints: e.g., have bounded curvature

  • Global constraints: e.g., have minimal length


Types of path constraints1

Types of Path Constraints

  • Local constraints: e.g., lie in free space

  • Differential constraints: e.g., have bounded curvature

  • Global constraints: e.g., have minimal length


Car like robot

f

q

f

Car-Like Robot

L

q

y

x

Configuration space is 3-dimensional: q = (x, y,q)


Example car like robot

f

dx/dt = v cosq

dy/dt = v sinq

dx sinq – dy cosq = 0

q

dq/dt = (v/L) tan f

f

|f| <F

Example: Car-Like Robot

L

q

y

x

Configuration space is 3-dimensional: q = (x, y,q)

But control space is 2-dimensional: (v, f) with |v| = sqrt[(dx/dt)2+(dy/dt)2]


Example car like robot1

f

dx/dt = v cosq

dy/dt = v sinq

dx sinq – dy cosq = 0

q

dq/dt = (v/L) tan f

f

|f| <F

Example: Car-Like Robot

L

q

y

x

Lower-bounded turning radius


How can this work tangent space velocity space

f

q

(x,y,q)

L

(dx,dy,dq)

q

q

f

y

x

y

dx/dt = v cosq

dy/dt = v sinq

(dx,dy)

x

q

dq/dt = (v/L) tan f

|f|<F

How Can This Work?Tangent Space/Velocity Space

dx sinq – dy cosq = 0


How can this work tangent space velocity space1

f

q

(x,y,q)

L

(dx,dy,dq)

q

q

f

y

x

y

dx/dt = v cosq

dy/dt = v sinq

(dx,dy)

x

q

dq/dt = (v/L) tan f

|f|<F

How Can This Work?Tangent Space/Velocity Space


Type 1 maneuver

CYL(x,y,dq,h)

h

h

dq

(x,y,q0)

  • = 2rtandq

    d = 2r(1/cosdq - 1) > 0

(x,y)

Type 1 Maneuver

q

r

dq

dq

 Allows sidewise motion

r

When dq 0, so does d and the cylinder becomes arbitrarily small


Type 2 maneuver

Type 2 Maneuver

 Allows pure rotation


Combination

Combination


Combination1

Combination


Coverage of a path by cylinders

+

q

q’

Coverage of a Path by Cylinders

q

y

x

Path created ignoring the car constraints


Path examples

Path Examples


Drawbacks

Drawbacks

  • Final path can be far from optimal

  • Not applicable to car that can only move forward (e.g., think of an airplane)


Reeds and shepp paths

Reeds and Shepp Paths


Reeds and shepp paths1

Reeds and Shepp Paths

CC|C0

CC|C

C|CS0C|C

Given any two configurations,the shortest RS paths betweenthem is also the shortest path


Example of generated path

Example of Generated Path

Holonomic

Nonholonomic


Other technique control based sampling

dx sinq – dy cosq = 0

dx/dt = v cos q

dy/dt = v sin q

dq/dt = (v/L) tan f

|f| <F

Other Technique: Control-Based Sampling

  • 1. Select a node m

  • 2. Pick v, f, and dt

  • 3. Integrate motion from m

  • new configuration


Other technique control based sampling1

Other Technique: Control-Based Sampling

Indexing array:A 3-D grid is placed over the configuration space. Each milestone falls into one cell of the grid. A maximum number of milestones is allowed in each cell (e.g., 2 or 3).

Asymptotic completeness:If a path exists, the planner is guaranteed to find one if the resolution of the grid is fine enough.


Computed paths

Computed Paths

Tractor-trailer

Car That Can Only Turn Left

jmax=45o, jmin=22.5o

jmax=45o


Application

Application


Architectural design verification of building code

Architectural Design: Verification of Building Code

C. Han


Other similar robots moving objects nonholonomic

Other “Similar” Robots/Moving Objects (Nonholonomic)

  • Rolling-with-no-sliding contact (friction), e.g.: car, bicycle, roller skate

  • Submarine, airplane

  • Conservation of angular momentum: satellite robot, under-actuated robot, catWhy is it useful?

    - Fewer actuators: design simplicity, less weight

    - Convenience (think about driving a car with 3 controls!)


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